Properties

Label 21.6.a.c
Level 21
Weight 6
Character orbit 21.a
Self dual Yes
Analytic conductor 3.368
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 21.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(3.36806021607\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 5q^{2} \) \(\mathstrut +\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut 7q^{4} \) \(\mathstrut +\mathstrut 94q^{5} \) \(\mathstrut +\mathstrut 45q^{6} \) \(\mathstrut -\mathstrut 49q^{7} \) \(\mathstrut -\mathstrut 195q^{8} \) \(\mathstrut +\mathstrut 81q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 5q^{2} \) \(\mathstrut +\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut 7q^{4} \) \(\mathstrut +\mathstrut 94q^{5} \) \(\mathstrut +\mathstrut 45q^{6} \) \(\mathstrut -\mathstrut 49q^{7} \) \(\mathstrut -\mathstrut 195q^{8} \) \(\mathstrut +\mathstrut 81q^{9} \) \(\mathstrut +\mathstrut 470q^{10} \) \(\mathstrut +\mathstrut 52q^{11} \) \(\mathstrut -\mathstrut 63q^{12} \) \(\mathstrut -\mathstrut 770q^{13} \) \(\mathstrut -\mathstrut 245q^{14} \) \(\mathstrut +\mathstrut 846q^{15} \) \(\mathstrut -\mathstrut 751q^{16} \) \(\mathstrut -\mathstrut 2022q^{17} \) \(\mathstrut +\mathstrut 405q^{18} \) \(\mathstrut +\mathstrut 1732q^{19} \) \(\mathstrut -\mathstrut 658q^{20} \) \(\mathstrut -\mathstrut 441q^{21} \) \(\mathstrut +\mathstrut 260q^{22} \) \(\mathstrut -\mathstrut 576q^{23} \) \(\mathstrut -\mathstrut 1755q^{24} \) \(\mathstrut +\mathstrut 5711q^{25} \) \(\mathstrut -\mathstrut 3850q^{26} \) \(\mathstrut +\mathstrut 729q^{27} \) \(\mathstrut +\mathstrut 343q^{28} \) \(\mathstrut +\mathstrut 5518q^{29} \) \(\mathstrut +\mathstrut 4230q^{30} \) \(\mathstrut +\mathstrut 6336q^{31} \) \(\mathstrut +\mathstrut 2485q^{32} \) \(\mathstrut +\mathstrut 468q^{33} \) \(\mathstrut -\mathstrut 10110q^{34} \) \(\mathstrut -\mathstrut 4606q^{35} \) \(\mathstrut -\mathstrut 567q^{36} \) \(\mathstrut -\mathstrut 7338q^{37} \) \(\mathstrut +\mathstrut 8660q^{38} \) \(\mathstrut -\mathstrut 6930q^{39} \) \(\mathstrut -\mathstrut 18330q^{40} \) \(\mathstrut -\mathstrut 3262q^{41} \) \(\mathstrut -\mathstrut 2205q^{42} \) \(\mathstrut +\mathstrut 5420q^{43} \) \(\mathstrut -\mathstrut 364q^{44} \) \(\mathstrut +\mathstrut 7614q^{45} \) \(\mathstrut -\mathstrut 2880q^{46} \) \(\mathstrut +\mathstrut 864q^{47} \) \(\mathstrut -\mathstrut 6759q^{48} \) \(\mathstrut +\mathstrut 2401q^{49} \) \(\mathstrut +\mathstrut 28555q^{50} \) \(\mathstrut -\mathstrut 18198q^{51} \) \(\mathstrut +\mathstrut 5390q^{52} \) \(\mathstrut +\mathstrut 4182q^{53} \) \(\mathstrut +\mathstrut 3645q^{54} \) \(\mathstrut +\mathstrut 4888q^{55} \) \(\mathstrut +\mathstrut 9555q^{56} \) \(\mathstrut +\mathstrut 15588q^{57} \) \(\mathstrut +\mathstrut 27590q^{58} \) \(\mathstrut -\mathstrut 11220q^{59} \) \(\mathstrut -\mathstrut 5922q^{60} \) \(\mathstrut -\mathstrut 45602q^{61} \) \(\mathstrut +\mathstrut 31680q^{62} \) \(\mathstrut -\mathstrut 3969q^{63} \) \(\mathstrut +\mathstrut 36457q^{64} \) \(\mathstrut -\mathstrut 72380q^{65} \) \(\mathstrut +\mathstrut 2340q^{66} \) \(\mathstrut +\mathstrut 1396q^{67} \) \(\mathstrut +\mathstrut 14154q^{68} \) \(\mathstrut -\mathstrut 5184q^{69} \) \(\mathstrut -\mathstrut 23030q^{70} \) \(\mathstrut +\mathstrut 18720q^{71} \) \(\mathstrut -\mathstrut 15795q^{72} \) \(\mathstrut +\mathstrut 46362q^{73} \) \(\mathstrut -\mathstrut 36690q^{74} \) \(\mathstrut +\mathstrut 51399q^{75} \) \(\mathstrut -\mathstrut 12124q^{76} \) \(\mathstrut -\mathstrut 2548q^{77} \) \(\mathstrut -\mathstrut 34650q^{78} \) \(\mathstrut +\mathstrut 97424q^{79} \) \(\mathstrut -\mathstrut 70594q^{80} \) \(\mathstrut +\mathstrut 6561q^{81} \) \(\mathstrut -\mathstrut 16310q^{82} \) \(\mathstrut -\mathstrut 81228q^{83} \) \(\mathstrut +\mathstrut 3087q^{84} \) \(\mathstrut -\mathstrut 190068q^{85} \) \(\mathstrut +\mathstrut 27100q^{86} \) \(\mathstrut +\mathstrut 49662q^{87} \) \(\mathstrut -\mathstrut 10140q^{88} \) \(\mathstrut -\mathstrut 3182q^{89} \) \(\mathstrut +\mathstrut 38070q^{90} \) \(\mathstrut +\mathstrut 37730q^{91} \) \(\mathstrut +\mathstrut 4032q^{92} \) \(\mathstrut +\mathstrut 57024q^{93} \) \(\mathstrut +\mathstrut 4320q^{94} \) \(\mathstrut +\mathstrut 162808q^{95} \) \(\mathstrut +\mathstrut 22365q^{96} \) \(\mathstrut +\mathstrut 4914q^{97} \) \(\mathstrut +\mathstrut 12005q^{98} \) \(\mathstrut +\mathstrut 4212q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
5.00000 9.00000 −7.00000 94.0000 45.0000 −49.0000 −195.000 81.0000 470.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) \(\mathstrut -\mathstrut 5 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(21))\).